{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0001","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=0 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"1","explanation_zh":"0!=1（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260927,"parameters":{"n":0,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":0,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0002","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=1 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"1","explanation_zh":"1!=1（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260928,"parameters":{"n":1,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":1,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0003","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=2 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"2","explanation_zh":"2!=2（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260929,"parameters":{"n":2,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":2,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0004","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=3 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"6","explanation_zh":"3!=6（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260930,"parameters":{"n":3,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":3,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0005","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=4 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"24","explanation_zh":"4!=24（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260931,"parameters":{"n":4,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":4,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0006","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=5 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"120","explanation_zh":"5!=120（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260932,"parameters":{"n":5,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":5,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0007","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=6 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"720","explanation_zh":"6!=720（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260933,"parameters":{"n":6,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":6,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0008","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=7 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"5040","explanation_zh":"7!=5040（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260934,"parameters":{"n":7,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":7,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0009","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=8 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"40320","explanation_zh":"8!=40320（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260935,"parameters":{"n":8,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":8,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0010","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=9 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"362880","explanation_zh":"9!=362880（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260936,"parameters":{"n":9,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":9,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0011","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=10 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"3628800","explanation_zh":"10!=3628800（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260937,"parameters":{"n":10,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":10,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0012","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=11 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"39916800","explanation_zh":"11!=39916800（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260938,"parameters":{"n":11,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":11,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0013","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=12 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"479001600","explanation_zh":"12!=479001600（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260939,"parameters":{"n":12,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":12,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0014","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=13 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"6227020800","explanation_zh":"13!=6227020800（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260940,"parameters":{"n":13,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":13,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0015","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=14 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"87178291200","explanation_zh":"14!=87178291200（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260941,"parameters":{"n":14,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":14,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0016","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=15 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"1307674368000","explanation_zh":"15!=1307674368000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260942,"parameters":{"n":15,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":15,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0017","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=16 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"20922789888000","explanation_zh":"16!=20922789888000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260943,"parameters":{"n":16,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":16,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0018","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=17 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"355687428096000","explanation_zh":"17!=355687428096000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260944,"parameters":{"n":17,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":17,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0019","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=18 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"6402373705728000","explanation_zh":"18!=6402373705728000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260945,"parameters":{"n":18,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":18,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0020","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=19 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"121645100408832000","explanation_zh":"19!=121645100408832000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260946,"parameters":{"n":19,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":19,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0021","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=20 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"2432902008176640000","explanation_zh":"20!=2432902008176640000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260947,"parameters":{"n":20,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":20,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0022","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=21 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"51090942171709440000","explanation_zh":"21!=51090942171709440000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260948,"parameters":{"n":21,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":21,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0023","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=22 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"1124000727777607680000","explanation_zh":"22!=1124000727777607680000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260949,"parameters":{"n":22,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":22,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0024","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=23 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"25852016738884976640000","explanation_zh":"23!=25852016738884976640000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260950,"parameters":{"n":23,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":23,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-exact-evaluation-0025","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘的精確值","stem_zh":"計算自然數 n=24 的 n!；答案以十進位整數字串表示。","answer":{"kind":"short_text","value":"620448401733239439360000","explanation_zh":"24!=620448401733239439360000（以 BigInt 精確計算；0!=1）。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘","BigInt"]},"variant":{"family_id":"qf-factorial-exact-evaluation","seed":20260951,"parameters":{"n":24,"operation":"exact_factorial","semantic_template":"qf-factorial-exact-evaluation/{\"n\":24,\"operation\":\"exact_factorial\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0001","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 0!=1。用 (n+1)!=(n+1)×n! 求 1!。","answer":{"kind":"short_text","value":"1","explanation_zh":"1×1=1，所以 1!=1。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261027,"parameters":{"n":0,"n_factorial":"1","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":0,\"n_factorial\":\"1\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0002","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 1!=1。用 (n+1)!=(n+1)×n! 求 2!。","answer":{"kind":"short_text","value":"2","explanation_zh":"2×1=2，所以 2!=2。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261028,"parameters":{"n":1,"n_factorial":"1","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0003","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 2!=2。用 (n+1)!=(n+1)×n! 求 3!。","answer":{"kind":"short_text","value":"6","explanation_zh":"3×2=6，所以 3!=6。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261029,"parameters":{"n":2,"n_factorial":"2","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0004","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 3!=6。用 (n+1)!=(n+1)×n! 求 4!。","answer":{"kind":"short_text","value":"24","explanation_zh":"4×6=24，所以 4!=24。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261030,"parameters":{"n":3,"n_factorial":"6","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0005","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 4!=24。用 (n+1)!=(n+1)×n! 求 5!。","answer":{"kind":"short_text","value":"120","explanation_zh":"5×24=120，所以 5!=120。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261031,"parameters":{"n":4,"n_factorial":"24","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0006","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 5!=120。用 (n+1)!=(n+1)×n! 求 6!。","answer":{"kind":"short_text","value":"720","explanation_zh":"6×120=720，所以 6!=720。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261032,"parameters":{"n":5,"n_factorial":"120","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0007","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 6!=720。用 (n+1)!=(n+1)×n! 求 7!。","answer":{"kind":"short_text","value":"5040","explanation_zh":"7×720=5040，所以 7!=5040。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261033,"parameters":{"n":6,"n_factorial":"720","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0008","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 7!=5040。用 (n+1)!=(n+1)×n! 求 8!。","answer":{"kind":"short_text","value":"40320","explanation_zh":"8×5040=40320，所以 8!=40320。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261034,"parameters":{"n":7,"n_factorial":"5040","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0009","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 8!=40320。用 (n+1)!=(n+1)×n! 求 9!。","answer":{"kind":"short_text","value":"362880","explanation_zh":"9×40320=362880，所以 9!=362880。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261035,"parameters":{"n":8,"n_factorial":"40320","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0010","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 9!=362880。用 (n+1)!=(n+1)×n! 求 10!。","answer":{"kind":"short_text","value":"3628800","explanation_zh":"10×362880=3628800，所以 10!=3628800。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261036,"parameters":{"n":9,"n_factorial":"362880","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0011","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 10!=3628800。用 (n+1)!=(n+1)×n! 求 11!。","answer":{"kind":"short_text","value":"39916800","explanation_zh":"11×3628800=39916800，所以 11!=39916800。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261037,"parameters":{"n":10,"n_factorial":"3628800","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0012","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 11!=39916800。用 (n+1)!=(n+1)×n! 求 12!。","answer":{"kind":"short_text","value":"479001600","explanation_zh":"12×39916800=479001600，所以 12!=479001600。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261038,"parameters":{"n":11,"n_factorial":"39916800","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0013","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 12!=479001600。用 (n+1)!=(n+1)×n! 求 13!。","answer":{"kind":"short_text","value":"6227020800","explanation_zh":"13×479001600=6227020800，所以 13!=6227020800。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261039,"parameters":{"n":12,"n_factorial":"479001600","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0014","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 13!=6227020800。用 (n+1)!=(n+1)×n! 求 14!。","answer":{"kind":"short_text","value":"87178291200","explanation_zh":"14×6227020800=87178291200，所以 14!=87178291200。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261040,"parameters":{"n":13,"n_factorial":"6227020800","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0015","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 14!=87178291200。用 (n+1)!=(n+1)×n! 求 15!。","answer":{"kind":"short_text","value":"1307674368000","explanation_zh":"15×87178291200=1307674368000，所以 15!=1307674368000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261041,"parameters":{"n":14,"n_factorial":"87178291200","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0016","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 15!=1307674368000。用 (n+1)!=(n+1)×n! 求 16!。","answer":{"kind":"short_text","value":"20922789888000","explanation_zh":"16×1307674368000=20922789888000，所以 16!=20922789888000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261042,"parameters":{"n":15,"n_factorial":"1307674368000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0017","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 16!=20922789888000。用 (n+1)!=(n+1)×n! 求 17!。","answer":{"kind":"short_text","value":"355687428096000","explanation_zh":"17×20922789888000=355687428096000，所以 17!=355687428096000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261043,"parameters":{"n":16,"n_factorial":"20922789888000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0018","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 17!=355687428096000。用 (n+1)!=(n+1)×n! 求 18!。","answer":{"kind":"short_text","value":"6402373705728000","explanation_zh":"18×355687428096000=6402373705728000，所以 18!=6402373705728000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261044,"parameters":{"n":17,"n_factorial":"355687428096000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0019","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 18!=6402373705728000。用 (n+1)!=(n+1)×n! 求 19!。","answer":{"kind":"short_text","value":"121645100408832000","explanation_zh":"19×6402373705728000=121645100408832000，所以 19!=121645100408832000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261045,"parameters":{"n":18,"n_factorial":"6402373705728000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0020","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 19!=121645100408832000。用 (n+1)!=(n+1)×n! 求 20!。","answer":{"kind":"short_text","value":"2432902008176640000","explanation_zh":"20×121645100408832000=2432902008176640000，所以 20!=2432902008176640000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261046,"parameters":{"n":19,"n_factorial":"121645100408832000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0021","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 20!=2432902008176640000。用 (n+1)!=(n+1)×n! 求 21!。","answer":{"kind":"short_text","value":"51090942171709440000","explanation_zh":"21×2432902008176640000=51090942171709440000，所以 21!=51090942171709440000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261047,"parameters":{"n":20,"n_factorial":"2432902008176640000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0022","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 21!=51090942171709440000。用 (n+1)!=(n+1)×n! 求 22!。","answer":{"kind":"short_text","value":"1124000727777607680000","explanation_zh":"22×51090942171709440000=1124000727777607680000，所以 22!=1124000727777607680000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261048,"parameters":{"n":21,"n_factorial":"51090942171709440000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0023","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 22!=1124000727777607680000。用 (n+1)!=(n+1)×n! 求 23!。","answer":{"kind":"short_text","value":"25852016738884976640000","explanation_zh":"23×1124000727777607680000=25852016738884976640000，所以 23!=25852016738884976640000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261049,"parameters":{"n":22,"n_factorial":"1124000727777607680000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0024","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 23!=25852016738884976640000。用 (n+1)!=(n+1)×n! 求 24!。","answer":{"kind":"short_text","value":"620448401733239439360000","explanation_zh":"24×25852016738884976640000=620448401733239439360000，所以 24!=620448401733239439360000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261050,"parameters":{"n":23,"n_factorial":"25852016738884976640000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-successor-recurrence-0025","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼階乘遞迴","stem_zh":"已知 24!=620448401733239439360000。用 (n+1)!=(n+1)×n! 求 25!。","answer":{"kind":"short_text","value":"15511210043330985984000000","explanation_zh":"25×620448401733239439360000=15511210043330985984000000，所以 25!=15511210043330985984000000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-successor-recurrence","seed":20261051,"parameters":{"n":24,"n_factorial":"620448401733239439360000","operation":"successor_recurrence","semantic_template":"qf-factorial-successor-recurrence/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"successor_recurrence\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0001","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 1!/0!；分母已明示為正。","answer":{"kind":"short_text","value":"1","explanation_zh":"1/1=1，因為 1!=1×0!=1。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261127,"parameters":{"n":1,"n_factorial":"1","previous_factorial":"1","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0002","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 2!/1!；分母已明示為正。","answer":{"kind":"short_text","value":"2","explanation_zh":"2/1=2，因為 2!=2×1!=2。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261128,"parameters":{"n":2,"n_factorial":"2","previous_factorial":"1","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0003","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 3!/2!；分母已明示為正。","answer":{"kind":"short_text","value":"3","explanation_zh":"6/2=3，因為 3!=3×2!=6。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261129,"parameters":{"n":3,"n_factorial":"6","previous_factorial":"2","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"2\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0004","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 4!/3!；分母已明示為正。","answer":{"kind":"short_text","value":"4","explanation_zh":"24/6=4，因為 4!=4×3!=24。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261130,"parameters":{"n":4,"n_factorial":"24","previous_factorial":"6","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0005","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 5!/4!；分母已明示為正。","answer":{"kind":"short_text","value":"5","explanation_zh":"120/24=5，因為 5!=5×4!=120。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261131,"parameters":{"n":5,"n_factorial":"120","previous_factorial":"24","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"24\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0006","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 6!/5!；分母已明示為正。","answer":{"kind":"short_text","value":"6","explanation_zh":"720/120=6，因為 6!=6×5!=720。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261132,"parameters":{"n":6,"n_factorial":"720","previous_factorial":"120","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"120\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0007","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 7!/6!；分母已明示為正。","answer":{"kind":"short_text","value":"7","explanation_zh":"5040/720=7，因為 7!=7×6!=5040。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261133,"parameters":{"n":7,"n_factorial":"5040","previous_factorial":"720","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"720\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0008","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 8!/7!；分母已明示為正。","answer":{"kind":"short_text","value":"8","explanation_zh":"40320/5040=8，因為 8!=8×7!=40320。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261134,"parameters":{"n":8,"n_factorial":"40320","previous_factorial":"5040","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"5040\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0009","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 9!/8!；分母已明示為正。","answer":{"kind":"short_text","value":"9","explanation_zh":"362880/40320=9，因為 9!=9×8!=362880。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261135,"parameters":{"n":9,"n_factorial":"362880","previous_factorial":"40320","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"40320\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0010","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 10!/9!；分母已明示為正。","answer":{"kind":"short_text","value":"10","explanation_zh":"3628800/362880=10，因為 10!=10×9!=3628800。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261136,"parameters":{"n":10,"n_factorial":"3628800","previous_factorial":"362880","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"362880\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0011","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 11!/10!；分母已明示為正。","answer":{"kind":"short_text","value":"11","explanation_zh":"39916800/3628800=11，因為 11!=11×10!=39916800。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261137,"parameters":{"n":11,"n_factorial":"39916800","previous_factorial":"3628800","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"3628800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0012","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 12!/11!；分母已明示為正。","answer":{"kind":"short_text","value":"12","explanation_zh":"479001600/39916800=12，因為 12!=12×11!=479001600。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261138,"parameters":{"n":12,"n_factorial":"479001600","previous_factorial":"39916800","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"39916800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0013","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 13!/12!；分母已明示為正。","answer":{"kind":"short_text","value":"13","explanation_zh":"6227020800/479001600=13，因為 13!=13×12!=6227020800。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261139,"parameters":{"n":13,"n_factorial":"6227020800","previous_factorial":"479001600","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"479001600\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0014","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 14!/13!；分母已明示為正。","answer":{"kind":"short_text","value":"14","explanation_zh":"87178291200/6227020800=14，因為 14!=14×13!=87178291200。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261140,"parameters":{"n":14,"n_factorial":"87178291200","previous_factorial":"6227020800","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6227020800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0015","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 15!/14!；分母已明示為正。","answer":{"kind":"short_text","value":"15","explanation_zh":"1307674368000/87178291200=15，因為 15!=15×14!=1307674368000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261141,"parameters":{"n":15,"n_factorial":"1307674368000","previous_factorial":"87178291200","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"87178291200\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0016","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 16!/15!；分母已明示為正。","answer":{"kind":"short_text","value":"16","explanation_zh":"20922789888000/1307674368000=16，因為 16!=16×15!=20922789888000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261142,"parameters":{"n":16,"n_factorial":"20922789888000","previous_factorial":"1307674368000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1307674368000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0017","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 17!/16!；分母已明示為正。","answer":{"kind":"short_text","value":"17","explanation_zh":"355687428096000/20922789888000=17，因為 17!=17×16!=355687428096000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261143,"parameters":{"n":17,"n_factorial":"355687428096000","previous_factorial":"20922789888000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"20922789888000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0018","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 18!/17!；分母已明示為正。","answer":{"kind":"short_text","value":"18","explanation_zh":"6402373705728000/355687428096000=18，因為 18!=18×17!=6402373705728000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261144,"parameters":{"n":18,"n_factorial":"6402373705728000","previous_factorial":"355687428096000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"355687428096000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0019","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 19!/18!；分母已明示為正。","answer":{"kind":"short_text","value":"19","explanation_zh":"121645100408832000/6402373705728000=19，因為 19!=19×18!=121645100408832000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261145,"parameters":{"n":19,"n_factorial":"121645100408832000","previous_factorial":"6402373705728000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"6402373705728000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0020","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 20!/19!；分母已明示為正。","answer":{"kind":"short_text","value":"20","explanation_zh":"2432902008176640000/121645100408832000=20，因為 20!=20×19!=2432902008176640000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261146,"parameters":{"n":20,"n_factorial":"2432902008176640000","previous_factorial":"121645100408832000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"121645100408832000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0021","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 21!/20!；分母已明示為正。","answer":{"kind":"short_text","value":"21","explanation_zh":"51090942171709440000/2432902008176640000=21，因為 21!=21×20!=51090942171709440000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261147,"parameters":{"n":21,"n_factorial":"51090942171709440000","previous_factorial":"2432902008176640000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"2432902008176640000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0022","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 22!/21!；分母已明示為正。","answer":{"kind":"short_text","value":"22","explanation_zh":"1124000727777607680000/51090942171709440000=22，因為 22!=22×21!=1124000727777607680000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261148,"parameters":{"n":22,"n_factorial":"1124000727777607680000","previous_factorial":"51090942171709440000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"51090942171709440000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0023","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 23!/22!；分母已明示為正。","answer":{"kind":"short_text","value":"23","explanation_zh":"25852016738884976640000/1124000727777607680000=23，因為 23!=23×22!=25852016738884976640000。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261149,"parameters":{"n":23,"n_factorial":"25852016738884976640000","previous_factorial":"1124000727777607680000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"1124000727777607680000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0024","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 24!/23!；分母已明示為正。","answer":{"kind":"short_text","value":"24","explanation_zh":"620448401733239439360000/25852016738884976640000=24，因為 24!=24×23!=620448401733239439360000。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261150,"parameters":{"n":24,"n_factorial":"620448401733239439360000","previous_factorial":"25852016738884976640000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"25852016738884976640000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-consecutive-quotient-0025","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"連續階乘的商","stem_zh":"計算 25!/24!；分母已明示為正。","answer":{"kind":"short_text","value":"25","explanation_zh":"15511210043330985984000000/620448401733239439360000=25，因為 25!=25×24!=15511210043330985984000000。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘商","正除數"]},"variant":{"family_id":"qf-factorial-consecutive-quotient","seed":20261151,"parameters":{"n":25,"n_factorial":"15511210043330985984000000","previous_factorial":"620448401733239439360000","operation":"consecutive_quotient","semantic_template":"qf-factorial-consecutive-quotient/{\"n\":25,\"n_factorial\":\"15511210043330985984000000\",\"operation\":\"consecutive_quotient\",\"previous_factorial\":\"620448401733239439360000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0001","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：1!=□×0!，其中 0!=1。","answer":{"kind":"short_text","value":"1","explanation_zh":"1= 1×1，所以 □=1。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261227,"parameters":{"n":0,"n_factorial":"1","successor_factorial":"1","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":0,\"n_factorial\":\"1\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0002","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：2!=□×1!，其中 1!=1。","answer":{"kind":"short_text","value":"2","explanation_zh":"2= 2×1，所以 □=2。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261228,"parameters":{"n":1,"n_factorial":"1","successor_factorial":"2","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":1,\"n_factorial\":\"1\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"2\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0003","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：3!=□×2!，其中 2!=2。","answer":{"kind":"short_text","value":"3","explanation_zh":"6= 3×2，所以 □=3。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261229,"parameters":{"n":2,"n_factorial":"2","successor_factorial":"6","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":2,\"n_factorial\":\"2\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0004","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：4!=□×3!，其中 3!=6。","answer":{"kind":"short_text","value":"4","explanation_zh":"24= 4×6，所以 □=4。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261230,"parameters":{"n":3,"n_factorial":"6","successor_factorial":"24","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":3,\"n_factorial\":\"6\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"24\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0005","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：5!=□×4!，其中 4!=24。","answer":{"kind":"short_text","value":"5","explanation_zh":"120= 5×24，所以 □=5。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261231,"parameters":{"n":4,"n_factorial":"24","successor_factorial":"120","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":4,\"n_factorial\":\"24\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"120\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0006","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：6!=□×5!，其中 5!=120。","answer":{"kind":"short_text","value":"6","explanation_zh":"720= 6×120，所以 □=6。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261232,"parameters":{"n":5,"n_factorial":"120","successor_factorial":"720","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":5,\"n_factorial\":\"120\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"720\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0007","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：7!=□×6!，其中 6!=720。","answer":{"kind":"short_text","value":"7","explanation_zh":"5040= 7×720，所以 □=7。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261233,"parameters":{"n":6,"n_factorial":"720","successor_factorial":"5040","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":6,\"n_factorial\":\"720\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"5040\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0008","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：8!=□×7!，其中 7!=5040。","answer":{"kind":"short_text","value":"8","explanation_zh":"40320= 8×5040，所以 □=8。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261234,"parameters":{"n":7,"n_factorial":"5040","successor_factorial":"40320","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":7,\"n_factorial\":\"5040\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"40320\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0009","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：9!=□×8!，其中 8!=40320。","answer":{"kind":"short_text","value":"9","explanation_zh":"362880= 9×40320，所以 □=9。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261235,"parameters":{"n":8,"n_factorial":"40320","successor_factorial":"362880","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":8,\"n_factorial\":\"40320\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"362880\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0010","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：10!=□×9!，其中 9!=362880。","answer":{"kind":"short_text","value":"10","explanation_zh":"3628800= 10×362880，所以 □=10。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261236,"parameters":{"n":9,"n_factorial":"362880","successor_factorial":"3628800","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":9,\"n_factorial\":\"362880\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"3628800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0011","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：11!=□×10!，其中 10!=3628800。","answer":{"kind":"short_text","value":"11","explanation_zh":"39916800= 11×3628800，所以 □=11。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261237,"parameters":{"n":10,"n_factorial":"3628800","successor_factorial":"39916800","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"39916800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0012","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：12!=□×11!，其中 11!=39916800。","answer":{"kind":"short_text","value":"12","explanation_zh":"479001600= 12×39916800，所以 □=12。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261238,"parameters":{"n":11,"n_factorial":"39916800","successor_factorial":"479001600","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":11,\"n_factorial\":\"39916800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"479001600\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0013","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：13!=□×12!，其中 12!=479001600。","answer":{"kind":"short_text","value":"13","explanation_zh":"6227020800= 13×479001600，所以 □=13。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261239,"parameters":{"n":12,"n_factorial":"479001600","successor_factorial":"6227020800","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":12,\"n_factorial\":\"479001600\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6227020800\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0014","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：14!=□×13!，其中 13!=6227020800。","answer":{"kind":"short_text","value":"14","explanation_zh":"87178291200= 14×6227020800，所以 □=14。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261240,"parameters":{"n":13,"n_factorial":"6227020800","successor_factorial":"87178291200","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":13,\"n_factorial\":\"6227020800\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"87178291200\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0015","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：15!=□×14!，其中 14!=87178291200。","answer":{"kind":"short_text","value":"15","explanation_zh":"1307674368000= 15×87178291200，所以 □=15。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261241,"parameters":{"n":14,"n_factorial":"87178291200","successor_factorial":"1307674368000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":14,\"n_factorial\":\"87178291200\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1307674368000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0016","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：16!=□×15!，其中 15!=1307674368000。","answer":{"kind":"short_text","value":"16","explanation_zh":"20922789888000= 16×1307674368000，所以 □=16。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261242,"parameters":{"n":15,"n_factorial":"1307674368000","successor_factorial":"20922789888000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":15,\"n_factorial\":\"1307674368000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"20922789888000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0017","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：17!=□×16!，其中 16!=20922789888000。","answer":{"kind":"short_text","value":"17","explanation_zh":"355687428096000= 17×20922789888000，所以 □=17。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261243,"parameters":{"n":16,"n_factorial":"20922789888000","successor_factorial":"355687428096000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":16,\"n_factorial\":\"20922789888000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"355687428096000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0018","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：18!=□×17!，其中 17!=355687428096000。","answer":{"kind":"short_text","value":"18","explanation_zh":"6402373705728000= 18×355687428096000，所以 □=18。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261244,"parameters":{"n":17,"n_factorial":"355687428096000","successor_factorial":"6402373705728000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":17,\"n_factorial\":\"355687428096000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"6402373705728000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0019","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：19!=□×18!，其中 18!=6402373705728000。","answer":{"kind":"short_text","value":"19","explanation_zh":"121645100408832000= 19×6402373705728000，所以 □=19。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261245,"parameters":{"n":18,"n_factorial":"6402373705728000","successor_factorial":"121645100408832000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":18,\"n_factorial\":\"6402373705728000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"121645100408832000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0020","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：20!=□×19!，其中 19!=121645100408832000。","answer":{"kind":"short_text","value":"20","explanation_zh":"2432902008176640000= 20×121645100408832000，所以 □=20。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261246,"parameters":{"n":19,"n_factorial":"121645100408832000","successor_factorial":"2432902008176640000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":19,\"n_factorial\":\"121645100408832000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"2432902008176640000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0021","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：21!=□×20!，其中 20!=2432902008176640000。","answer":{"kind":"short_text","value":"21","explanation_zh":"51090942171709440000= 21×2432902008176640000，所以 □=21。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261247,"parameters":{"n":20,"n_factorial":"2432902008176640000","successor_factorial":"51090942171709440000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":20,\"n_factorial\":\"2432902008176640000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"51090942171709440000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0022","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：22!=□×21!，其中 21!=51090942171709440000。","answer":{"kind":"short_text","value":"22","explanation_zh":"1124000727777607680000= 22×51090942171709440000，所以 □=22。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261248,"parameters":{"n":21,"n_factorial":"51090942171709440000","successor_factorial":"1124000727777607680000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":21,\"n_factorial\":\"51090942171709440000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"1124000727777607680000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0023","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：23!=□×22!，其中 22!=1124000727777607680000。","answer":{"kind":"short_text","value":"23","explanation_zh":"25852016738884976640000= 23×1124000727777607680000，所以 □=23。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261249,"parameters":{"n":22,"n_factorial":"1124000727777607680000","successor_factorial":"25852016738884976640000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":22,\"n_factorial\":\"1124000727777607680000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"25852016738884976640000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0024","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：24!=□×23!，其中 23!=25852016738884976640000。","answer":{"kind":"short_text","value":"24","explanation_zh":"620448401733239439360000= 24×25852016738884976640000，所以 □=24。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261250,"parameters":{"n":23,"n_factorial":"25852016738884976640000","successor_factorial":"620448401733239439360000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":23,\"n_factorial\":\"25852016738884976640000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"620448401733239439360000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260926-factorial-missing-successor-factor-0025","batch_id":"batch-20260926-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"後繼式的缺失因子","stem_zh":"填入整數 □：25!=□×24!，其中 24!=620448401733239439360000。","answer":{"kind":"short_text","value":"25","explanation_zh":"15511210043330985984000000= 25×620448401733239439360000，所以 □=25。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["階乘遞迴","BigInt"]},"variant":{"family_id":"qf-factorial-missing-successor-factor","seed":20261251,"parameters":{"n":24,"n_factorial":"620448401733239439360000","successor_factorial":"15511210043330985984000000","operation":"missing_successor_factor","semantic_template":"qf-factorial-missing-successor-factor/{\"n\":24,\"n_factorial\":\"620448401733239439360000\",\"operation\":\"missing_successor_factor\",\"successor_factorial\":\"15511210043330985984000000\"}"}},"verification":{"status":"candidate","checks":["schema_pending","independent_recomputation_pending"],"warning_zh":"本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-09-26","contract_version":"ocme-question-factory-v0.1"}}